Converting between bases

Lesson

Denary to binary conversion

1. Create a grid:

128	64	32	16	8	4	2	1

2. Add a 1 to the corresponding cell if number contributes to target number and 0 to all the other cells

Worked example: convert 24₁₀ to binary.

128	64	32	16	8	4	2	1
0	0	0	1	1	0	0	0

16₁₀+8₁₀=24₁₀

The binary value is 11000₂ (we can ignore the preceding zeros)

Binary to denary conversion

Worked example: Convert 01011001₂ to denary

1. Create the grid:

128	64	32	16	8	4	2	1
0	1	0	1	1	0	0	1

2. Add up the cells that have a corresponding value of 1:

64 + 16₁₀ + 8₁₀ + 1= 89₁₀

Hexadecimal to denary conversion

1. Convert the two hex values separately to denary value
2. Multiply the first value by 16
3. Add the second value

Worked example: Covert A3₁₆ to denary

A₁₆ = 10₁₀

3₁₆ = 3₁₀

(10₁₀ x 16₁₀) +3₁₀ = 163₁₀

Denary to hexadecimal conversion

1. Integer divide the denary number by 16
2. Take the modulus 16 of the denary number
3. Convert the two numbers to the corresponding hex values.

Worked example: Covert 189₁₀ to hex

189₁₀ / 16₁₀ = 11₁₀ remainder 15₁₀

11₁₀ = B₁₆

15₁₀ = F₁₆

189₁₀ = BF₁₆

Hexadecimal to binary conversion

1. Find the corresponding 4-bit binary number for the two numbers
2. Concatenate the two binary values to give the final binary value

Worked example: Covert C3₁₆ to binary

C₁₆ = 12₁₀ = 1100₂

3₁₆= 3₁₀ = 0011₂

11000011₂

Binary to hexadecimal conversion

1. Split the binary number into groups of 4 bits: 1110₂ 1010₂
2. Find the corresponding Hex value for each of the 4-bit groups

Worked example: Covert 11101010₂ to hexadecimal

1110₂ |1010₂

1110₂ = 14₁₀ = E₁₆

1010₂ = 10₁₀ = A₁₆

EA₁₆

Learning Videos

For more information click on the tab below to watch a video about the lesson.

Click for video - Converting between decimal and 8 bit binary

Click for video - Converting between decimal and 2 digit hexadecimal

Questions

1. For each of the binary values below, write down the decimal equivalent.

   a) 1101

   b) 1111

   c) 00100110

   d) 10110111

   
   

   a) 13

   b) 15

   c) 38

   d) 183

2. For each of the decimal values below, write down the binary equivalent.

   a) 18

   b) 57

   c) 163

   d) 255

   
   

   a) 00010010

   b) 00111001

   c) 10100011

   d) 11111111

   
   

   When converting from denary to binary, leading zeros do not need to be written. So (a) would be just as correct to write 10010. However, if the question asks for an 8 bit number, then only 00010010 would be correct.

3. Circle the binary value below that represents the decimal value 87.

   (a) 01011010

   (b) 11000100

   (c) 01010111

   (d) 00011010

   
   

   (c) 01010111

4. Decimal to Hexadecimal conversion

   (i) Convert decimal 19 to hexadecimal:

   (ii) Convert decimal 44 to hexadecimal:

   (iii) Convert hexadecimal 19 to decimal:

   (iv) Convert hexadecimal A3 to decimal:

   
   

   i) 13

   ii) 2C

   iii) 25

   iv) 163

5. Binary to hexadecimal conversion:

   (i) Convert binary 00110101 to hexadecimal:

   (ii) Convert binary 11010111 to hexadecimal:

   (iii) Convert hexadecimal 1E to binary:

   (iv) Convert hexadecimal FF to binary:

   
   

   i) 35

   ii) D7

   iii) 0001 1110

   iv) 1111 1111

6. Convert the following values to and from Hexadecimal: (Show your working).

   (a) Decimal 37 to Hex

   (b) Decimal 59 to Hex

   (c) Hex 11 to Decimal

   (d) Hex 2F to Decimal

   (e) Hex 1A to Binary

   (f) Hex 16 to Binary

   (g) Binary 0011 0111 to Hex

   (h) Binary 1101 1111 to Hex

   
   

   a) 25

   b) 3B

   c) 17

   d) 47

   e) 0001 1010

   f) 0001 0110

   g) 37

   h) DF